Variable-order fractional Schrödinger equation

As a natural generalization of the fractional Schrödinger equation, the variable-order fractional Schrödinger equation has been exploited to study fractional quantum phenomena [1]

i\hbar {\frac {\partial \psi ^{\alpha (\mathbf {r} )}(\mathbf {r} ,t)}{\partial t^{\alpha (\mathbf {r} )}}}=(-\hbar ^{2}\Delta )^{\frac {\beta (t)}{2}}\psi (\mathbf {r} ,t)+V(\mathbf {r} ,t)\psi (\mathbf {r} ,t).

where $Δ = \partial 2 / \partial r 2$ is the Laplace operator and the operator (−ħ²Δ)^{β (t)/2} is the variable-order fractional quantum Riesz derivative.

References

A. Bhrawy and M. Zaky, " An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations”, Applied Numerical Mathematics, Volume 111, January 2017, Pages 197–218

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[1] A. Bhrawy and M. Zaky, " An improved collocation method for multi-dimensional space-time variable-order fractional Schrödinger equations”, Applied Numerical Mathematics, Volume 111, January 2017, Pages 197–218